TOPICS IN THE FOUNDATIONS OF GEOMETRY AND LOGIC

The objective of the thesis is the investigation of three distinct but related topics in the foundations of logic and geometry. We first present a development of the theory of affine constructible sets. Utilizing methods of logic and field theoretic algebraic geometry most of the known theorems in this subject are reproven and some further results are developed. We next study the foundations of projective geometry. The fundamental configurations of linear projective geometry are investigated and certain new logical relations between these configurations are demonstrated. The final topic treated in the thesis is the theory of universal classes with the amalgamation property. These classes are generalizations of the class of fields of some fixed characteristic and consequently we prove three generalizations of theorems of algebraic geometry in this context. In addition we present some extensions of results of first order logic to these classes and prove related results that the peculiar to this theory.